I bumped into papers interested in the following characters $$\chi_m(n) = \left\{ \begin{array}{cl} \left( \frac{m}{n} \right) & \text{if } m \equiv 1 \mod 4 \\ \left( \frac{4m}{n} \right) & \text{if } m \equiv 2,3 \mod 4 \\ \end{array} \right\}$$
where the symbols are the Kronecker symbols mod $m$ and $4m$ respectively. Why are these characters of particular interest and why do they arise naturally?
On
Let $m$ be a square-free integer and consider the quadratic extension $K=\mathbb{Q}(\sqrt{m})$ over $\mathbb{Q}$. We may define a Dirichlet character $$ \chi_K: \Big(\mathbb{Z} \big/ |\Delta_K|\mathbb{Z} \Big)^\times \longrightarrow \big\{ \pm1 \big\} $$ as you discribed in your question. here $\Delta_K = m$ if $m \equiv 1 \pmod{4}$ and $\Delta_K = 4m$ if $m \equiv 2,3 \pmod{4}$.
This character will play important role in many propositions and theorems about the quadratic field $K$.
For example, the decomposition of $p$ in $K$:
a prime number $p$ splits in $K$ $\Longleftrightarrow$ $\chi_K(p)=1$, $p$ is inert in $K$ $\Longleftrightarrow$ $\chi_K(p)=-1$ and $p$ is ramified in $K$ $\Longleftrightarrow$ $\chi_K(p)=0$.
And another example, the class number formula of $K$:
let $h_K$ be the class number of $K$.
if $K$ is imaginary and $K \neq \mathbb{Q}(\sqrt{-1}),\mathbb{Q}(\sqrt{-3})$, then $$ h_K = \frac1{2-\chi_K(2)} \left| \sum_{1 \leq k < \frac{1}{2}|\Delta_K|} \chi_K(k) \right|. $$ if $K$ is real, say $\varepsilon$ is a fundamental unit, then $$ h_K = \frac1{\log \varepsilon} \left| \sum_{1 \leq k < \frac12|\Delta_K|} \chi_K(k) \log\sin \frac{\pi k}{|\Delta_K|} \right|. $$
Quadratic Dirichlet characters arise naturally for reciprocity in number theory, i.e., for Gauss sums and Gauss quadratic reciprocity law, but also in $L$-functions and many other topics. As an example, the proof of Dirchlet's theorem on primes in arithmetic progressions uses such $L$-functions. A detailed and explicit account is given inTom Apostol's book "Introduction to analytic number theory".